G.151 circumscribe a triangle about a circle equiangular with a triangle — Heath

IV.3

About a given circle to circumscribe a triangle equiangular with a given triangle.

Let ( A B C ) be the given circle, and < D E F > the given triangle; thus it is required to circumscribe about the circle ( A B C ) a triangle equiangular with the triangle < D E F >.

Let / E F / be produced in both directions to the points G, H, let the centre K of the circle ( A B C ) be taken [G.98], and let the straight line / K B / be drawn across at random; on the straight line / K B /, and at the point K on it, let the angle < B K A ) be constructed equal to the angle < D E G ), and the angle < B K C ) equal to the angle < D F H ); [G.48] and through the points A, B, C let / L A M /, / M B N /, / N C L / be drawn touching the circle ( A B C ). [G.118]

Now, since / L M /, / M N /, / N L / touch the circle ( A B C ) at the points A, B, C, and / K A /, / K B /, / K C / have been joined from the centre K to the points A, B, C, therefore the angles at the points A, B, C are right. [G.120]

And, since the four angles of the quadrilateral < A M B K > are equal to four right angles, inasmuch as < A M B K > is in fact divisible into two triangles,

and the angles < K A M ), < K B M ) are right,

therefore the remaining angles < A K B ), < A M B ) are equal to two right angles.

But the angles < D E G ), < D E F ) are also equal to two right angles; [G.37] therefore the angles < A K B ), < A M B ) are equal to the angles < D E G ), < D E F ), of which the angle < A K B ) is equal to the angle < D E G );

therefore the angle < A M B ) which remains is equal to the angle < D E F ) which remains.

Similarly it can be proved that the angle < L N B ) is also equal to the angle < D F E );

therefore the remaining angle < M L N ) is equal to the angle < E D F ). [G.59]

Therefore the triangle < L M N > is equiangular with the triangle < D E F >; and it has been circumscribed about the circle ( A B C ).

Therefore about a given circle there has been circumscribed a triangle equiangular with the given triangle.

Q. E. F.

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